Is it permissible for my friend to return to work in her home country with her new nationality, while informing employers of that, or is it obligatory for her to resign?

That which governs the rights, duties, and employment conditions between an employee and the employer is the employment contract concluded between them, for Allah Almighty says: "O you who have believed, fulfill [all] contracts," and the Hadith (definition) states: "Muslims are bound by their stipulations." Therefore, regarding the female employee, it can only be said that she must adhere to the terms and conditions of the contract.